Asset Pricing under Asymmetric Information Strategic Market Order Models

Transcription

1 Kyle Asset Pricing under Asymmetric Strategic Market Order Models Markus K. Brunnermeier Princeton University August 17, 2007

2 Kyle A of Market Microstructure Models simultaneous submission of demand schedules competitive rational expectation models strategic share auctions sequential move models screening models in which the market maker submits a supply schedule first static uniform price setting limit order book analysis dynamic sequential trade models with multiple trading rounds strategic market order models where the market maker sets prices ex-post

3 Strategic Market Order Models - Overview Kyle model static version dynamic version (in discrete time) Refresher in continuous time version (Back 1992) Multi-insider version Other strategic market order models

4 Kyle 1985 Model Kyle Model Setup asset return v N (p 0, Σ 0 ) Agents (risk neutral) Insider who knows v and submit market order of size x Noise trader who submit market orders of exogenous aggregate size u N (0, σ 2 u) Market maker sets competitive price after observing net order flow X = x + u Timing (order of moves) Stage 1: Insider & liquidity traders submit market orders Stage 2: Market Maker sets the execution price Repeated trading in dynamic version

5 Kyle 1985 Model Version Kyle Single informed trader (Competitive) Market Maker 0) 0) v := asset s payoff X = x + u batch net order flow 1) Conjecture (price-rule) 1) Conjecture (insider trading rule) p = µ + λ(x + u) x = α + βv 2) No Updating 2) Updating E[v x + u] 3) Optimal Demand 3) Price Setting Rule max x E[(v p) v]x p = E[v x + u] max x E[v µ λx v]x p = E[v] + Cov[v,x+u] Var[x+u] {x + u E[x + u]} FOC: x = µ 2λ + 1 2λ v p = p 0 + βσ0 β 2 Σ 0+σ {x + u α + βe[v]} u 2 SOC: λ > 0 4) Correct Beliefs 4) Correct Beleifs α = µ 2λ, β = 1 2λ µ = p 0 Martingale, λ = βσ0 β 2 Σ 0+σu 2

6 Kyle 1985 Model Version Kyle solve for unknown coefficients 4 unknown Greeks 4 equations λ = 1 2 Σ0 σ 2 u λ (illiquidity) decreases with noise trading, σ 2 u Σ 0 reflect info advantage of insider

7 Kyle - A Refresher The Problem: max for several periods t = 1,..., T (discrete time) [ T ] max E t v s ( x s, u s ) t (sequential rationality) u t s=1 under the following law of motion x t+1 = f t ( x t, u t, ε t ) x t : vector of state variables (sufficient state space) u t : vector of control variables ε t : vector of random shocks Method Backward Induction

8 Kyle - A Refresher Define Value Function T V t (x t ) := E t [ v s ( x s, us ) ] s=t optimal values Bellman Equation Start at final date T max u t E t [v t (x t, u t ) + V t+1 (x t+1 )] V T +1 ( ) := 0 in t = T max u T E T [v T (x T, u T )] FOC u T = g T (x T ) V T (x T ) = E T [v T (x T, u T )]

9 Kyle - A Refresher at date T 1 max u T 1,u T E T 1 given V T (x T ) [ T s=t 1 v s ( x s, u s ) max ut 1 E T 1 [v T 1 (x T 1, u T 1 ) + V T (x T )] given law of motion max ut 1 E T 1 [v T 1 (x T 1, u T 1 ) + V T (f T 1 (x T 1, u T 1, ε T 1 ))] ut 1 = g T 1 (x T 1 ) V T 1 = [ ( ( ( E T 1 vt 1 xt 1, ut 1) + VT ft 1 xt 1, ut 1, ε ))] T 1 and so on for date T 2 etc. (and if they didn t die in the uncertainties they are still solving...) This process is quite time consuming. ]

10 Kyle - A Refresher Alternative way: Step 1: Guess the general form of the value function V t+1 (x t+1 ) = H t+1 (x t+1 ) }{{} e.g. H t+1 (x t+1 )=α t+1 x 2 t+1 Step 2: Derive optimal level of current control max u t E t [v t (x t, u t ) + H t+1 ( x t+1 )] max u t E t [v t (x t, u t ) + H t+1 (f t (x t, u t, ε t ))] u t = Step 3: Derive value function and check whether it coincides with general value function V t (x t ) = E t [v t (x t, u t ) + H t+1 (f t (x t, u t, ε t ))]? = H t (x t ) = α t x 2 t

11 Kyle Insider Version Step 1: Conjectured price setting strategy (pricing rule) p n = p n 1 + λ n X n = p n 1 + λ n ( x n + u n ) ( ) 1 = Liquidity λ t Step 2: Guess Value function for insider s profit pricing rule is linear guess quadratic value function) E[π n+1 p 1,, p n, v }{{} set up to n ] = α n (v p n ) 2 + δ n (expected profit from time n + 1 onwards) [ π n = E n πn+1 + (v p n ) xn i ]

12 Kyle Version Insider ctd. Step 3: Write Bellman Equation max E (v p n ) x i xn i n + α n (v p n ) 2 + δ n p 1,, p n 1, v }{{} I n 1 Step 4: Given insider s beliefs p n = p n 1 + λ n X n [ ( v pn 1 λ n x i ) ] n λ n u n x i max E ( n xn i +α n v pn 1 λ n xn i ) 2 λ n u n + δn I n Take expectations ( v pn 1 λ n xn i max x i n ) x i n + α n ( v p n 1 λ n xn i }{{} state variable t n +δ n + α n λ 2 n σ 2 u }{{} u p n noisy }{{} control ) 2

13 Kyle Insider ctd. Step 5: maximize Version FOC: (v p n 1 ) 2λ n x i n 2α n λ n (v p n 1 )+2α n λ 2 n x i n = xn i = 1 2α nλ n (v p n 1 ) 2λ n (1 α n λ n ) }{{} :=β n t n SOC: λ n (1 α n λ n ) > 0 Step 6: Check whether guessed value fcn is correct E [π I n 1 ] = max E xn i [ (v p n ) x i n + α n (ṽ p n ) 2 + δ n I n 1 = α n 1 (v p n 1 ) 2 + δ n 1, where α n 1 = 1 4λ n (1 α n λ n ), δ n 1 = δ n + α n (λ n ) 2 σu t 2 n

14 Kyle Version Market Maker (Filtering Problem) Step 1: MM s belief about insider s strategy x i n = β n t n (v p n 1 ) X n = β n t n (v p n 1 ) + u }{{} n Var[ u n]=σu 2 tn Step 2: MM s filtering problem By definition: p n 1 : = E [v X 1,, X n 1 ] Σ n 1 : = Var [v X 1,, X n 1 ] E [ X n X 1,, X n 1 ] = β n t n E [(v p n 1 ) + u n Var [ X n ] = (β n t n ) 2 Σ n 1 + σu t 2 n Cov [v, X n ] = E [v (β n t n (v p n 1 ) + u n = β n t n Σ n 1

15 Kyle Version Now we have all ingredients for the Projection Theorem n p n = p n 1 + β n t n Σ n 1 (β n t n ) 2 X n Σ n 1 + tσu }{{ 2 } :=λ n (β n t n ) 2 Σ 2 n 1 = V [v X n ] = Σ n 1 (β n t n ) 2 Σ n 1 + tσu 2 = σ 2 uσ n 1 (β n ) 2 t n Σ n 1 + σ 2 u λ n = n β n t n n 1 (β n t n ) 2 Σ n 1 + tσ 2 u = (1 λ n β n t n ) Σ n 1 = σ2 uλ n β n λ n = β n Σ n /σ 2 u

16 Version Kyle Step 3: Equate coefficients α n,β n,δ n, n β n t n = 1 2αnλn 2λ n(1 α n) 1 α n 1 = 4λ n(1 α nλ n) Solve recursive δ n 1 = δ n + α n λ 2 nσu t 2 n system of n = } σ2 uσ n 1 equations. as above λ n =... Interpretation of Equilibrium restrain from aggressive trading price impact in current trading round price impact in all future trading rounds...

17 Generalizations of Kyle Multiple Insiders all have same information all hold different information information is correlated see Foster & Viswanathan, JF 51, Risk averse insiders CARA utility etc. etc.